Fig. 3: The simulation results of transmitting a high-dimensional state at the single-photon level.

a, b The correct result of \(\left| \psi \right\rangle _b\), conditional on \(\left| {\psi ^{0,0}} \right\rangle _{ac}\). c, d The incorrect result of \(U_{2,0}^ + \left| \psi \right\rangle _b\), conditional on \(\left| {\psi ^{2,0}} \right\rangle _{ac}\); e, f the incorrect result of \(U_{10,10}^ + \left| \psi \right\rangle _b\), conditional on \(\left| {\psi ^{10,10}} \right\rangle _{ac}\). Left and right columns show the spectra, \(a_{\ell ,p}\) and \(\phi _{\ell ,p}\), respectively, while the trivial background of a maximally mixed state \(\frac{1}{d}\mathop {\sum}\nolimits_j {\left| j \right\rangle } \left\langle j \right|\) is not shown